THE FRACTAL DIMENSION OF THE
WEIERSTRASS TYPE FUNCTIONS
by
..-'
LEE TI^N WAH
•
Thesis
Submitted to the Faculty of the Graduate School of
The Chinese University of Hong Kong
(Division of Mathematics)
In partial fulfillment of the requirements
for the Degree of
Master of Philosophy
July, 1998 f^ ^S&-.•<:"^^^^¾!!^^^^^ _ £^^
H \ 9 JULJ^HI k "uN!VER^"""""“^/^I ^^SUBRARY SYST^^ 'X^^^ 撮要
本論文的目的是計算一些與Weierstrass函數有關的 圖形的分形維數0 Weierstrass函數是一個眾所周知 連續但不可微分的函數。關於這個函數的圖形， 我們知道它的盒維數但是它的Hausdorff維數則仍 是一個公開而具有挑戰性的問題。在論文中，我 們運用兩種方法來計算Weierstrass函數圖形的 Hausdorff維數，他們是質量分佈準則和位勢理論方 法。首先我們構造一個量度並運用質量分佈準則 來計算它的Hausdorff維數的下限。然後我們加入 一個參數9在?616『31『355函數裏並透過位勢理論方 法，則對幾乎所有參數e我們可以得出它的 Hausdorff維數。自仿射曲線是另一種圖形由迭代 函數系統構造而成。這種圖形可分為兩種類別。 第一種是可微分的自仿射曲線，他們是易於計算 的。另一種是分形的自仿射曲線，它們是我們主 要關注的課題。正如Weierstrass函數，它的 Hausdorff維數仍不容易計算。我們運用位勢理論 方法來計算它的Hausdorff維數。最後，我們引進 Dekking[ll]的 recurrent M � Recurrent 集的表示法是將 代數和分析聯系起來，並且是一種有效的方法來 描述和製造space-filling 圖形和一些相似於 Weierstrass函數的圖形。 The Fractal dimension of the Weierstrass type functions 1 Abstract
Our aim in the thesis is to estimate the fractal dimension of certain graphs related to the Weierstrass functions. The Weierstrass function is the best-known example of a continuous but nowhere differentiable function. The box dimension of the graph is well-known, however, its Hausdorff dimension is a still challenging open problem. In this thesis, two methods are used to investigate the Hausdorff dimen- sion of the Weierstrass function[5]. They are the mass distribution principle[l and the potential theoretic method[l]. By using the mass distribution principle, a measure is constructed to estimate the lower bound ofthe Hausdorff dimension[5 .
The potential theoretic method introduces a new parameter and gives simple for- mula to express the Hausdorff dimension for，，almost all" cases in the parameter space[7]. Self-affine curves are constructed by Iterated function system (IFS) and are another type of curves analogous to the Weierstrass function. We discuss the regularity of such curves. The differentiable case is easy. For the fractal case, the
Hausdorff dimension is difficult to obtain. Falconer[3] use the potential theoretic method to study such curve and its Hausdorff dimension. Finally, the recurrent set[ll] is introduced which is primarily based on interplaying between algebra and analysis. Its description is a powerful method of describing and generating space-filling curves and graphs of nowhere differentiable continuous functions like the Weierstrass function. The Fractal dimension of the Weierstrass type functions 2
ACKNOWLEDGMENTS
I wish to express my sincere and deepest gratitude to my supervisor, Profes- sor K. S. Lau, for introducing this topic to me. With his inspired guidance and discussions and achieve the thesis, I have increased my knowledge in this frontier of mathematics.
LEE TIN WAH
The Chinese University of Hong Kong
August 1998 Contents
1 Introduction 5
2 Preliminaries 8
2.1 Box dimension and Hausdorff dimension 8
2.2 Basic properties of dimensions 9
2.3 Calculating dimensions 11
3 Dimension of graph of the Weierstrass function 14
3.1 Calculating dimensions of a graph 14
3.2 Weierstrass function 16
3.3 An almost everywhere argument 23
3.4 Tagaki function 26
4 Self-afRne mappings 30
4.1 Box dimension of self-affine curves 30
4.2 Differentability of self-affine curves 35
4.3 Tagaki function 42
4.4 Hausdorff dimension of self-affine sets 43
3 The Fractal dimension of the Weierstrass type functions 4
5 Recurrent set and Weierstrass-like functions 56
5.1 Recurrent curves 56
5.2 Recurrent sets 62
5.3 Weierstrass-like functions from recurrent sets 64
Bibliography Chapter 1
Introduction
The focus of the thesis is to discuss a variety of interesting fractals occuring
as graphs of functions. The usual way of measuring a fractal is by some form
of 'dimension'. The ,dimension, refers to the box dimension and Hausdorff di-
mension. The former one is conceptually simple and easy to use. Its popularity
is largely due to its relative ease of mathematical calculation and empirical es-
timation. However, the latter one is more desirable mathematically dues to the
property of countable stability. But it is harder to get the lower estimates. There
are two techniques to finding the Hausdorff dimension of a fractal set. They are
the mass distribution principle[l] and the potential theoretic method[l]. These
techniques are used repeatedly in this thesis.
The Weierstrass function 00 f{x) = J2 X—aj sin(27rA^) (1.1) j=i
is the most famous example of a continuous but nowhere differentiable function.
Weierstrass proved that this function is nowhere differentiable for some of these
values of A and a, while Hardy [6] gave the first proof for all such A and a. The
dimension of the graph ofthe function is suggested to be 2 — a. The box dimension
was proved to be the case (under some mild condition)[l], but it remains open
5 The Fractal dimension of the Weierstrass type functions 6 whether the Hausdorff dimension of the graph to have the same value. In Chapter
2 Mass distribution principle [1] is used to show that the Hausdorff dimension of the graph of the Weierstrass function is very closed to 2 — a where A is large[5 .
This technique is also used in chapter 3 to estimate the Hausdorff dimension of the graph of Tagaki function[2], a class of resemble functions. In another consideration, we let H — [0,1]°° endowed with the uniform probability measure,
and let 0 — {〜，Oi, • • .}• We introduce a parameter to the Weierstrass function oo fe{x) 二 E A—^_ sin(27rA^ + Oj). 3 = l
By showing that the t-energy of fjLe,
/tM = / / (Or-")2 + (/,(�— /,("))2)t/2
is finite for t < 2 — a, Hunt[7] concludes that the Hausdorff dimension of the
graph of fe over [0，1] is 2 — a for almost every 0 G H dues to the potential
theoretic method where /i^ is induced by Lebesgue measure C on [0,1 .
Self-affine sets form another important class of fractal sets. An affine map
S : R^ ~^ EP is a transformation of the form
S{x) = T{x) + b
where T is a linear transformation on R" (representable by an n x n matrix) and b is a vector in R^. Suppose each Si {i = 1，...，m) is a contracting affine contractions. Then by a theorem of Hutchinson [8] there is a unique compact invariant set F with the property that
F = USi{F).
We call F a self-affine set. Like the Weierstrass function, the Hausdorff dimension of the self-affine set is difficult to obtain. Potential theoretic method[l] is used to show that for almost The Fractal dimension of the Weierstrass type functions 7 all bi there is a formula d{Ti, •.. ’ T^) expressing the HausdorfF dimension where d{Ti, • •., Tm) is a function depended on the singular value of T] [3 .
Then, self-affine sets in the plane can also be considered as curves of func- tions by suitable choice of affine transformations. Given a collection of points
(xo, Vo), • • •，{xm, ym)- Let Si be the affine transformation represented in matrix notation with respect to {x, y) coordinates by
(X \ ( 1/m 0 \ ( X \ ( {i — l)/m� Si — + • \ y ) \ «i Ci } \ y J \ bi
The properties Si{{xo,yo)) = {xi,yi) and Si[{xm,ym)) = (^m+i,Wi) are hold to ensure that the self-affine set is a curve. In the thesis we discuss the cases that the self-affine curves are differentiable or fractal type. The box dimension for the fractal type is l + log(CiH hc^)/logm[9]. Falconer[2] proves that the formula d(Ti, • • • ,Tm) has the same value in this case. Also the self-affine curve of the difFerentable case is studied on some special condition.
Finally, recurrent set[ll] is introduced which is primarily based on an inter-
play between algebra and analysis. The recurrent sets are the limits of a sequence
of compact subsets of R^ associated with iterates of a given word under an ap-
propriate free semigroup endomorphism. Its description is a powerful method of
describing and generating space-filling curves, graphs of nowhere differentiable
continuous functions. Chapter 2
Preliminaries
2.1 Box dimension and Hausdorff dimension
Box dimension is one of the most widely used dimensions. It is easy to cal- culate and is widely used by the other scientists, even thought it lacks many desirable mathematical properties. For a non-empty bounded subset E, Let
E\ 二 sup{|o; — y\ : X, y G E} be the diameter of a set E C M^. Let Nr{E)
be the smallest number of sets of diameter r that can cover E. The lower and
upper box dimensions of E are defined as
dimB^ = liminflog， r—O — log r
厂 ^ .. logNr(E) dim^jC/ = lim sup r^^0 - log r respectively. If these two expressions are equal, we refer to the common value as
the box dimension of E, i.e., .^ 1. logjVrQg) dimsE 二 lim ^^. r^0 — logr Thus the least number of sets of diameter r which can cover E is roughly of order r_s where s 二 dims^. The box dimension is unchanged if the covering set of E is replaced by closed balls, cubes, r-mesh cubes.
8 The Fractal dimension of the Weierstrass type functions 9
On the other hand, the Hausdorff dimension is defined in terms of a measure and is more sophisticated than the box dimension. Suppose that F is a subset of
M" and s is a non-negative number. For any 6 > 0, we define ‘oo 飞 n's{F) = inf Y1 \Ui\' : {Ui} is a ^-cover of F \ (2.1) .i=l J As S decreases, the class of permissible covers of F in (2.1) is reduced. Therefore, the infimum %l{F) increases and so approaches a limit as 5 ~> 0. We write
W[F) = limm[F). “0
We call W[F) the s-dimensional Hausdorff measure of F. It is easy to show that for all sets F C M" there is a number dim^F, called the Hausdorff dimension of
F, such that n'{F) = oo if s < dim^F and n'{F) = 0 if s > dim^F. Thus
dimnF 二 inf{s : W{F) = 0} - sup{s : W{F) - oo}.
The Hausdorff dimension of a set F may be thought of as the number s at which
W{F) 'jumps，from oo to 0. When s = diuinF, the measure W{F) can be zero or infinite, and if 0 < W {F) < oo, F is called an s-set.
2.2 Basic properties of dimensions
It is important to understand the relationship between box dimension and
Hausdorff dimension. For each F C E^, it can be covered by Ns{F) sets of
diameter ¢^, it follows from definition (2.1) that,
n|{F) < Ns{F)S^
For each s < dim^F, we have W{F) > e > 0. So logiV^(F) + slog6 > loge if
6 is sufficiently small. Thus, s < liminfj_^o logN5{F)/(- log(5) and this implies
that diuiH F < dim^F < ^sF. (2.2) The Fractal dimension of the Weierstrass type functions 10
The following properties hold with ‘ dim' denoting any of Hausdorff or box di- mension.
(i) If Fi C F2, then dimFi < dimF2.
(ii) If F is finite, then dimF = 0.
(iii) If F is a non-empty open subsets of E^, then dimF = n.
(iv) If F is a smooth m-dimensional manifold in E^, then dimF = m.
(v) If f : F "> W is bi-Lipschitz, i.e.
b\x - y\ < \f{x) - f{y)l < a\x — y| (工，y G F)
where 0 < b < a < 00, then dim/(F) = dimF.
(vi) If f is a similarity or affine transformation, then dim/(F) = dimF.
Hausdorff dimension is countably stable, that is dim (UgiFj) = supi However, box dimension is finitely stable but not countably stable. A simple ex- ample is that F = {0,1,1/2,1/3,.. •} is a countable set with dim^ F = |. Count- able stability is one of the main advantages of Hausdorff dimensions over box dimensions; in particular it implies that countable sets have Hausdorff dimension zero. We also recall that dim^F = dim^F and dim^F 二 dim^F where F is the closure of F. In fact this is a disadvantage of box dimensions, since we often wish to study a fractal F that is dense in an open region of R^ and which therefore has box dimension n. The Fractal dimension of the Weierstrass type functions 11 2.3 Calculating dimensions We frequently wish to estimate the Hausdorff dimensions of sets; usually it is easy to estimate the upper bound by constructing a special cover. To obtain a lower bound we must show that ^ |^7^|^ is greater than some positive constant for all ^-coverings of F. There are two usual approaches to finding the dimension of a set. They are the "mass distribution principle"[1] and the "potential theoretic method"[l]. The former is a method that involve studying a suitable measure supported by the set and the latter concerns with a density argument. Theorem 2.3.1 (Mass distribution principle) Let F C R" and let fj, be a finite measure with fj,{F) > 0. Suppose that there are s > 0； c > 0 and S > 0 such that _ < c\U\' for all sets U with \U\ < 6. Then W{F) > fJ.{F)/c and s < dimiyF < dim^F < dim^F. Proof If {Ui] is any cover of F then / \ 0 Taking infimum, we have %l{F) < /x(F)/c if 5 is small enough, so that W{F) > "(F)/c. • A useful modification of the above is Proposition 2.3.2 Let F C M^ be a Borel set, let jjL be a finite Borel measure on R^ and 0 < c < oo. (i) //limsupr—o"(BO^，r))/rS < c for all x e F, then W[F) > A^(F)/c. (ii) If limsup^_,o f^{B{x, r))/r' > c for all x G F, then W{F) < 2'^i{F)/c. The Fractal dimension of the Weierstrass type functions 12 Note that it is immediate from Proposition 2.3.2 that dimn F =linv—o log/x(5y.(x))/logr if this limit exists. Potential theoretic method[l] is another technique for calculating Hausdorff dimensions that is important both in theory and in practice. This replaces the need for estimating the mass of a large number of small sets by a single check for the convergence of a certain integral. For s > 0, the s-energy of mass distribution ^i on W is =f f d^{x)d^{y) J J |a:-ys The following theorem relates Hausdorff dimension to the seemingly unconnected potential theoretic ideas. In particular, if there is a mass distribution on a set F which has finite s-energy, then F has dimension at least s. Theorem 2.3.3 (Potential theoretic method) Let F be a subset of R^. If there is a mass distribution |j, on F with /s(^0 < ⑴，then 7i^{F) = oo and diniif F > s. Proof Suppose that /s(yU) < oo for some mass distribution jjL with support contained in F. Define Fi = \ X G F : limsnpfi{Br{x))/r' >ol. L r~>0 J We claim that /i(Fi) 二 0. If x G Fi, for any e > 0 there exists a sequence {r^} decreasing to 0 such that jjL{Br{x)) > erf. Also /i has no point mass, otherwise Is{|ji) = oo. By the continuity of /i, we can construct a sequence {qi) from {r^} such that jJi{Ai) > |er^-(z = 1, 2,...), where Ai is the annulus B^ (x) \Bq. {x). Now we get a subsequences {ri.} from {rJ such that ?^+i < qi- for all j so that the Ai are disjoint annuli centred on x. We define ^.w 二 [ ^ J |i-y|s The Fractal dimension of the Weierstrass type functions 13 for X e Fi. We have ^.(.) = J[ |^y|s^ � ^ f dfi{y) ~ fe^v l^-^l' 00 1 ^ Ertf=m j=i since \x — y\ < ri. on Ai-. However, /s(") = f ^s{^)d^i{x) < 00, we have 0s(x) < 00 for /^-almost all x. We conclude that yu(Fi) = 0 and prove the claim. The definition of Fi shows that limsup^_^o fJ'{^r{x))/r^ = 0 if x G F \ Fi. Applying Proposition 2.3.2(a) on F \ Fi, we have W{F) > W{F \ Fi) > "(F \ Fi)/c > (M(F) - /x(Fi))/c 二 M(F)/c. for all c > 0. Hence 9{'(F) = 00. • This theorem will be used in the following chapter to estimate the generic Hausdorff dimension of the graph of a family of Weierstrass functions. The tech- nique is to construct a measure and to prove that its energy is finite. Chapter 3 Dimension of graph of the Weierstrass function 3.1 Calculating dimensions of a graph Let f be a real valued function defined on [a, 6] and let Gr(/) = {(t, f{t)) : a < t < b} be the graph of /. If f has a continuous derivative, then it is not difficult to see that Gr(/) has dimension 1. The same is true if f is of bound variation. However, it is possible for a continuous function to be sufficiently irregular to have a graph of dimension strictly greater than 1. In the following we will derive some interesting estimate for the box and Hausdorff dimension of the graphs. Given a function f and an interval [ti, t2], we write Rf for the maximum range of f over an interval, Rf[tut2]^ sup lf{u)-f{v)l (3.1) t1 and m is the least integer greater than or equal to l/S. IfN5 is the number of squares of the 6 — mesh that intersect Gr{f), then m—l m—1 (5—1 Y^ Rf[z(^, {i + l)S] < Ns < 2m + 5_i ^ R{[i6, {i + l)S]. (3.2) i=0 i=0 14 The Fractal dimension of the Weierstrass type functions 15 Proof Since f is a continuous function, the number of mesh squares of side S in the column above the interval [iJ, {i + l)S] that intersect Gr(/) is at least Rf[iS, {i + l)5]/5 and at most 2+Rf[i(5, {i + l)S]/S. Summing over all much inter- vals gives (3.2). • In the following theorem, this proposition is applied to estimate the dimension of the graph of a function that satisfies the Holder condition. Theorem 3.1.2 Let f : [0,1] ~> R be a continuous function. (i) Suppose \f{u) - f{v)l < c\u - v\' (0 < u, V < 1) (3.3) where c� 0and 0 < s < 1. Then V?~'{F) < oo and dim/, Gr(/) < dims Gr(f) < 2 - s. This remains trues if (3.3) holds when \u — v\ < 5 for some S > 0. (ii) Suppose that there are numbers c > 0； (¾ > 0 and 1 < s < 2 with the following property: for each u € [0，1] and 0 < S < (¾; there exists v such that u —叫 S ^ and |/M-/WI�s . (3.4) Then 2 — s < dim^ Gr(/). Proof (i) Follows immediately from (3.3) that Rf[ti,^2] < c\ti — t2Y for 0 < t1,t2 < 1. With the same notations as in Proposition 3.1.1, we have m < 1 + S~^ so that Ns < (1 + ri)(2 + cS-^5') < ciS'-^, where Ci is independent of S. It follows from the definition that dim^ Gr(/) < 2 — s. We also see that 7^^(Gr(/)) < Ns{V2Sf-^ < V¥^c, < 00 The Fractal dimension of the Weierstrass type functions 16 for a sequence of values of S tending to 0. Thus dim/y(Gr(/)) < 2 — s. (ii) In the same way, (3.4) implies that Rf[t1,t2] > c\ti - t2|s. Since 6~^ < m, we have from (3.2) that m—l Ns > S-^ Y1 Rf[iS, {i + 1)5] > 6-^S-^c6' = c5'-^ i=0 so the definition gives dim^ Gr(/) > 2 — s. • 3.2 Weierstrass function Perhaps the best-known example of a continuous but nowhere differentiable function is the Weierstrass function 00 f{x) = Y^ X-aj sin(27rA^) (3.5) j=i where 0 < a < 1 and A > 1. The function has a colorful history and continue to make an appearance in various fields. Weierstrass proved that this function is nowhere differentiable for some values of A and a, 2| 1 1 1 1 1 1 1 1 1 ：她〜, ."\fV : _2l I I I I I I I I 1——I ::0 0.1 0.2 ^0.3 0.4 0.5 0.6w 0.7 0.8 0.9 1 Figure 1: Graph of f(x) with A = 2 and a — 0.5 The Fractal dimension of the Weierstrass type functions 17 while Hardy [6] gave the first proof for all such A and a. The graph of the Weier- strass function have often been studied as an important example of fractal curves. Its box dimension will be proved to be 2 - a[l] by Proposition 3.2.1 for A > 1. In Figure 1 we take A = 2 and a — 0.5, the box dimension is 1.5. It has been conjec- tured that the Hausdorff dimension of the graph of f also has the same value and the question is still unsolved. Note that dimn Gr(/) < dim^ Gr(/) = 2 - a from (2.2). In this section, the main effort is to show dimn Gr(/) is bounded below by 2 — a — C| ln6 [5] for some constants C independent of b. when b is large, the dimension is closed to 2 — a as in the conjecture. The method is to construct a special measure on a subset of graph of the Weierstrass function and apply the “mass distribution principle" [1] which has been introduced in Chapter 2. Proposition 3.2.1 Let f be the Weierstrass function in (3.5). Then, provided A is large enough, dim^ Gr(/) = 2 — a. Proof Given 0 < h < 1，let N be the integer such that A—(�+i <) h < A—�. Then, by using the mean-value theorem on the first N terms of the sum and an obvious estimate on the remainder. We have N |/(t + /i)-/(t)| < Y^ A""^| sin(27rA^(^ + h)) - sm{2nXH)\ k=l oo + Y1 A"^^| sin(27rA^(t + h)) - sin(27rA^t)| k=N+l N oo < ^27rA-"^A^/i+ Y1 2A-"^ k=l k=N+l 27T/iA(-^" 2入-啊1) - 1 — A«-1 + l-A-" < ch^ where c is independent of h. Theorem 3.1.2(i) implies that dim^ Gr(/) < 2 — a. For S < 7rA"\ take N such that 7rA"^ < 6 < 7rA_("-i). For each t, we may choose The Fractal dimension of the Weierstrass type functions 18 h with 7rA-("+i) < h < 7rA"^ such that | sin(27rA^(t + h)) — sin(27rA^t)| > |. If we split the sum into three parts: the first N — 1 terms, the iVth term, and the rest, we have, as the above estimate, \-aN 2\-aN |/(f+/0-/W-A-a"(sin(2iA"(f+/0)-sin(2iA"_ < ^TZ^ + 3^ (3.6) Suppose A > 2 is large enough for the last sum of (3.6) to be less than ^X~^^ for all N, then lf{t + h)-f{t)l > A"^^| sin(27rA^(t + h)) - sin(27rA^t)| (^ , 2 \ N VAi-«-l A«-l； > -A""^ > -X-^S'^ -4 4 It follows from Theorem 3.1.2(ii) that dim^ Gr(/) > 2 — a. • We now come to the main estimation. Theorem 3.2.2 Let f be the Weierstrass function in (3.5). Then there exists a constant C > 0 such that if X > 15，then dimHGT{f)>2-a-^. The theorem follows from the next lemma and mass distribution principle[l . Lemma 3.2.3 Let f be the Weierstrass function in (3.5). Then there is a con- stant C > 0，a constant Ci depends on A with X > 15，and a Cantor subset K in E together with a probability measure v supported on Gr(f) D (K x R) such that if X is a square of side h < A_i with sides parallel to the coordinate axes, then i^{X) < Ci/i(2-+i& Proof We divide the proof into 4 steps. Step 1. Construction of a Cantor subset K =门二丄 U|a|=n ^<^ ^^ 股 We assume I 二 [0, |] and r — [|] — 1. Since A > 15, r > 2. We define a system The Fractal dimension of the Weierstrass type functions 19 of intervals {J^| a G U^ii?n}, where Rn = {1,.. -,r}^, as follows. For each i, 1 < i < r, let Qi be the largest integer in the interval X{I + {i — 1)). Then, the integer qi — r is also in the interval. Set J _ \qi-r qi 」卜 L^T'lJ, then, Ji C I + {i — 1), i = 1,. • •, r and there exists e > 0 such that sin'(27rx) > e if X G Ji. Suppose Jcr has been chosen of the form J^ — [{qa- — r)/A'^', Q'a-/Al^'], where q^ is a positive integer, and |a| denotes the length of the sequence a. For each i e Rn, let Qa*i be the largest integer in the interval A(/ + gv — r^-i — 1). Set j — Qa*i — r Q,cr*i J� *i= [ AH+1，Ai^ . 1 1 1 1 1 1 1- r\ 厂 / \ sin(2*pi*x) / D.5_Z \ / • / \ sin(2*pi*15*x)/sqrt(15) / 4mk^Mjfk J11 J12 \ / J21 J22 _1- __ v^ __ - J1 J2 I 1 1 1 1 I 0 0.2 0.4 0.6 0.8 1 1.2 X It demonstrates an illustration of J�o nthe first 2 levels for A = 15. The definition gives that r = 2,gi = 3,g2 = 18. So, Ji = [j^, ^], J2 -[错,转]，Ji\ =[盖,盖]’ Ji2 =[盖，•]，^21 = |"241 243] „„J 7 _ rlM 1^1 l225' Il5J ana J22 — L225' 225J. Figure 2: Construction of interval J^ with A = 15 on first 2 level. The Fractal dimension of the Weierstrass type functions 20 Then, we check that J^ have the following properties: (J1) {Ja\cf G {1, • • •, r}^} is a collection of pairwise disjoint intervals of length rA_" for each n. (J2) For each a and i — 1, •.., r, Ja.i C A+l(/ + w-r + 0 \q^ - r + i - 1 gg-r + i1 C [ )^ ，AH J C Ja. (J3) Uxe Ja, then sin'(27rAl^l-^) = 2ir cos(27rAl^l"^) > e. Step 2. Constructing of a measure v Let 00 ^=n u Jo， n=l |o"|=n let fjL be the unique probability measure supported on K defined by the condition /i(J^) 二 r-丨“丨，and let v be the probability measure supported on Gr(/|^) defined by / g{x,y)dv{x,y) := / g{xJ{x))dfj.{x), (3.7) JR^ JR for g G C*o(^2). Let X be a square [xo,x^ + h] x [yo,yo + h] with h < A~^. Let n be the positive integer such that A_("+i) < h < A—�an dlet k be the positive integer such that A""(^+^) < h < 广"(_&-1). Let #s = card{a| |a| = n + s and Gr(/|j。）A X / ^} for each s G {0,1, •. •，k — 1}. It can be checked that "po = [ ix(t,m)Mt) JK < #(A:-l)r-(—-i). Step 3. Estimating #(A: - 1) < 2(5&-iA(i—#*—i) Suppose |cr| = p = n + s and Gr(/|j^) n X ^ 0. Let p-i ^(a;) = ^A-"^sin(27rA^). A:=0 The Fractal dimension of the Weierstrass type functions 21 We have � \-ap ||/_^|| - X~^P ^—ap _ g{x) - i — A-a，"(z) + i-x-a n[yo,2^ + "]/0 (3.8) If X G Ja, then from (J3) , we have P-1 ( \ l-ap _ 1 \ g,M = 27T [ A(1-W cos(27rA^) > 6 ^^^z^ • k=0 \ , In particular, g is increasing on J�Let E = {x 6 Ja\ equation (3.8) holds }. Since g is continuous and increasing on J^, E is an interval. If Gr(/|j。“）nX + ¢, then J for i = 1, • • •, r. Since these last intervals are nonoverlapping, E can meet at most 2 + X^C{E) of them , where L is Lebesgue measure, So, card{i e {1,. • •, r}\fU^, n X • 0} < 2 + X^L{E) (3.9) Consequently, #(s + U < card{ie{l,.-.,r}|/|j_nXM#s < 2 + An+s£(E)#s, It follows that JO[E) < •，where d = mm{g'{x)\x G J^} and c is the height of the box that g must be in if f is in X on J^. Now if (x, f{x)) G X, then (x, g{x)) is in a box of height h + 2A_^/(1 - A"^). Since h < 入-++&-1) ^nd p < n + k - 1, we have h < A"^^ < 广哗/(1 - A""). So, wr^、 3 ( A-哗 \ ( Ai—"-l \ 乙(丑）<:(1^) [w^l)- The Fractal dimension of the Weierstrass type functions 22 and hence 3/_^X ( Ai-M \ 十 1[I^X^J {xH-)p-lJ = Ai-{x^ + Kr^)G^^)}. (3.10) Let hp{X) be the factor {} in (3.10). Note that for all p > l,hp{X) < hi{X) for A > 15. Now, hi{X) is continuous and liniA^oo hi{X) = |. Let S = max{/ii(A)|A > 15}. Then we have | < S < oo. Therefore, from (3.9), we obtain #(s + l) < 5\ils. (3.11) Since #0 < 2, we find by recursion on (3.11), #(A;_l)S2#-iA(i-^^i). Step 4. Estimating of z/(X) From (3.7), we have p[X) < 2^-iA(i-#bi)r-(^*-i). Since n(l - a)/a < k < n{l - a)/a + 1 + l/a and r_i < 10/A. Thus, iy{X) < 2#J#-W/nO*+*-i)A-a(A-i)A-n < 2(5去5几(1_—/0 10时(明_—+1)7"入—"(冲_〜,<^-1/<^)入_几 < 2(lO0*r=[(lO&-")i/"rA-W-a). Set A = 2(10(5)i/a and B = (lOp-”"�Sinc e\h > A—^ p{X) < AX-^B^h^-^. Finally, since n < -ln h/ lnA, B^ 二 e"ins < "-inB/inA Set C = ln5. Let Ci 二 AA-a. We have V{X) < C7i"2-a-CyinA. To see that C > 0, it suffices to show that 10^^~" > 1. Since sine function has norm 1 and e/5 < 2. Since S > f, we have lOJi—� >3^""10". Thus, lOJ^"^ > 1, for 0 < a < 1. • The Fractal dimension of the Weierstrass type functions 23 3.3 An almost everywhere argument We now consider the Weierstrass function with a random phase added to each term : 00 fe{x) = Y^ \-^^ sin(27rA^ + Oj) (3.12) j=o where 0 = (¾, Oi,. •.). In [7], Hunt give the Hausdorff dimension of the graph of function defined in (3.12) is 2 — a for almost all 0. Theorem 3.3.1 If each On is chosen independently with respect to the uniform probability measure in [0,1], then with probability one on 0, the Hausdorff dimen- sion of the graph of fe is 2 — a. Proof Let H = [0,1]°° endowed with the uniform probability measure, and let 0 = (^o, ^i, • • •) denote a point in H. The upper bound of the Hausdorff dimension is obtain in Section 3.1. We now turn to the proof that the Hausdorff dimension of the graph of fe over [0,1] is at least 2 — a for almost every 0 G H. The proof is based on the potential-theoretic 'energy' approach to Hausdorff dimension which is introduced in Chapter 2. Let jjLe be the measure supported on the graph of fo that is induced by the Lebesgue measure C on [0，1]. That is, for X C E^, MX) = C{{x e [0，1] : (x, fe{x)) G X}). Then, the s-energy of fjie is 1八叫)=/o,i]/o,i] {{x-yy^{fo{x)-fe{y)n^'' We will show that the integral Es = f Is{^ie)dO JH is finite for s G (1，2-a), then Is(jie) is finite for almost every 9 G H if s < 2 — a, which implies that the Hausdorff dimension of the graph of fe is at least s. The Fractal dimension of the Weierstrass type functions 24 Choosing a sequence of values of s approaching 2 — a, we conclude that for almost every 6 G H, the Hausdorff dimension of the graph of fe is at least 2 — a. By the Fubini's theorem, Es = /o,i] /o,i] L {{x-yy^{fe{x)-fe{y)n^'^^^^' We claim that there exists C > 0 such that for 0 < \x — y\ < l/(2A^), f ^ Since s < 2 — a, it follows that Eg < oo. To prove the claim we fix x and y with 0 < \x — y\ < l/(2A^), and let z 二 /权⑷—fe{y). Regarding Z as a function of the random sequence G, we have Z is a random variable and will show that Z has a bounded density function h{z). For z — \x — y\w , we have r d^ _ 广 h{z)dz JHi{^-yy + {fe{x)-fe{y)n/' 二 loo((T1)2 + y)"2 / � h{\x — y\w)\x — y\dw -oo |x-7/Kl+^2)V2 < (sup"(20)|:r-"|i-s/ ^2W2. z J-oo [i + w^y/^ Now to complete the proof we only need to show that h{z) < Ci\x — y\~^ for some Ci > 0 that is independent of x and y. Note that, ^ 二 fe(oo) — fe{y) 00 =Y^ A-"^ (sin(27r(A^ + 6j)) - sin(27r(A^ + Oj))) j=o 00 =^ 2A-^^(sin(27rA^(x - y)/2) cos(27r(A^(x + y)/2 + 6j)) j=o 00 =Y^qjcos{rj + 27r0j) j=o where qj and Vj do not depend on 6 for all j. Let Zj = qj cos{rj + 27r^); then Zo, Zi, •.. are independent random variables (since ^o, ^i, • • • are independent ran- The Fractal dimension of the Weierstrass type functions 25 dom variables) with density functions M^.)4^丨她丨 [ 0 \zj\ > \qj (since rj + 2irQj is uniformly distributed on an interval of length 27r ). It follows that the density function h[z) for Z = Zo + Zi + • • • is the infinite convolution ho * hi * • •.. Since the maximum value of a probability density cannot increase under convolution with another probability density, any upper bound we obtain on a finite convolution hj * •.. * hk is an upper bound on h{z) as well. Next, recall that qj = 2A_w' sin(27rA^(a; - y)/2) and that 0 < \x - y\ < l/(2A^). Let k > 2 be the integer for which (l/2)A—“i < x-y\ < (l/2)A-*. Then, 7T ^ 、t> oX — y ^ .X — y 7T —< 27rA^-2-^ < 27rA^-^ < -, 2A3 2 2 - 2' and hence y>2sin(^)A->2-(^)|.-,r for j = k — 2, k — 1, k. Let || • ||p denote the U norm, and notice that hj G L^ for p < 2. It follows that for j = k — 2, k — 1, k, ||/lj3/2 = iq&||_l/3$i^l|:r-2/|-a/3 where K is an absolute constant and Ki depends only on A. By Young's inequal- ity, \hk-i * hkWs < ||"fc-1||3/2||Afc||3/2, and then by Holder's inequality, hk-2^hk-l*hk{z) < Whk-2W3/2Whk+l*hkW3 < ||"fc-2||3/2||^t-l||3/2||^t||3/2 < Kl\x-y\~^ Then, h(z) < Kf\x — y|"", and the proof is complete. • The Fractal dimension of the Weierstrass type functions 26 3.4 Tagaki function While it is difficult to obtain the exact Hausdorff dimension of the Weierstrass function, we try to replace the sine functions in (3.5) by a simpler function. In the following, we let g be the 'zig-zag' function of period 1 defined on R by f 2x (0 < X < 1/2) g{k + x) � < l-2x (1/2 < x < 1) V where k is an integer. To define a function oo f{x)^Y.XT-g{\x) (3.13) i=l for some special sequence {A^} where A > 1 and 0 < A < 1. We call this function Tagaki function [2]. The next theorem shows that the Hausdorff dimension of the graph of such function is 2 — a for some particular choice of {Aj)[2 . Theorem 3.4.1 Let f be the Tagaki function defined in (3.13). Suppose that {Aj} is a sequence of positive numbers with Aj+i/Aj increasing to infinity and logAi+i/ logAj ^ 1. Then dimjy Gr(/) = 2 — a. Proof If A^^i < h < Ap, then for h sufficiently small, k oo |/(x + /i)-/(x)| < J2X7-\g{Xi{x + h))-g{\x)\ + 2 ^ A-" i=l i=A;+l k oo < "EV^+i + 2E^-a i—l i=k+l < 2/a,-�+ +i 4A,-� < 2/i" + 4/i^-6/i". We conclude from Theorem 3.1.2 that dim^ Gr(/) < 2 - a. The lower estimate of the dimension of Gr(/) is more difficult to obtain. Let S be a square with sides of length h and parallel to the coordinate axes. The Fractal dimension of the Weierstrass type functions 27 Let I be the interval of projection of S onto the x-axis. We define a measure "(*S) = £({x : (x, f{x)) G S}) where C is the Lebesgue measure. We now show that u{S) < C/z2-a We let E = {x : (x, f(x)) e 5} and let k A(x) = ^Ar-^(A,x). i=i Furthermore, we assume that k is large enough to ensure that A^；+! > 2A^ > 2. Then 00 00 |/Or) — Mx)\ = I [ Afp(A#)| < [ A-" < 2A,7i (3.14) iz=k+l i=k+l and k k 1 \m\ -iEA-^v(A.x)i > A,-^i -x>r+i > Jv^+i i=l i=l except for countably many x. Suppose that the square S has side h = X^^ for some such k. Let m be the integer such that A,7^ < h = A,-i < A,7^_,. (3.15) Certainly, m > 1. On the other hand, since Xi+i/Xi is increasing, f ^k+l\ (m l)a \a z i �+爪 -1 �+ l\ � “— xa . A ^T" ^k < T ��� I — \+m-l < Afc, V Afc / \^k+m-2 ^k J So that / \ \ (m_l)a / X \ \ -oc+l / X \ {k-l){-a+l) (¾?) <、‘<(点..务）<(^r) A， Hence, on taking logarithms, m < ak where a is independent of k. We will consider three case, m = 1, 2 and greater than 2 separately: If m = 1, then by (3.14) (x, f{x)) can lie in S only if (a;, fk{x)) lies in the rectangle Si obtained by extending S a distance 2Af_^i < 2h above and below. The derivative /(� changes sign, at most, once in the interval I. On each section on which f'k{x) is of constant sign \fk{x)\ > \\\~^ so {x, fk{x)) can lie in Si for X in an interval of length, at most, 2入广"times the height of Si. Thus u{S) < 2 . 2A^+a • 5h = 20"2-a. The Fractal dimension of the Weierstrass type functions 28 If m = 2, we can divide I into, at most, two parts, on each of which f'k{x) is of constant sign. The height of Si is h + 4A^"^ < 5A^^^ by (3.15), so in each part we need only consider x belonging to a subinterval of length 2A^~^ • 5A^"^ when seeking points of E. We divide each of these subintervals further into parts on which /(+“a;) is also of constant sign. In this way we obtain, at most, 2A^~^ . 5入工• |Afc+i + 1 < 6 (^^) new intervals from each of the old one. If m > 2, we repeat the process to obtain from each of the last set of /入 \ -a+l intervals, at most, 6 |^-^j intervals on each of which fk+2{^) is also of constant sign. Proceeding in this way we eventually see that E is covered by, at most, / \ \ \ -oc+l /� \ _oH"l 0 nm-l / ^fc+l ^k+m-l 0 am-1 ^k+m-l 2-6 I ^ T =2.() "~"X~~ \ ^k ^k+m-2/ \ ^k / intervals on each of which /“爪](^;) is of constant sign. It follows that on each such interval, {x, f{x)) G S only if (x, fk^rn-i{^)) ^ ^¾, where S2 is the rectangle formed by extending S a distance 2Aj^"^ < 5h, so, by considering the gradient of fk+m-i on each such interval, we have / \ \ -ot+i K^) < 2.6-ip^ 善2入二_1 \ ^k < 20 • 6爪-1"2-" < 20 • 6a*"2-a Thus there exist constants b and c such that u{S) < ch^h?~^ if h = A^� Now suppose that S is a square of side h where A^^^ < h < Ap. It follows from the above that if t < 2 — a, \t Lk "�=" �< ch'>?r = cA,-: =),2 ,2. \ \ Hence, v[S)=^C\E) < cih\ (3.16) since A^2-a+t)/2 increases faster than A“i and A^2_a_^ increases faster than b^ for large k, in view of the stated growth conditions on A^. The Fractal dimension of the Weierstrass type functions 29 If {Ui} is any cover of Gr(/) enclose each U{ in a square Si of side equals Ui\. Writing Ei = {x : {x, f{x)) G 5J, we must have that [0,1] C UiEi. Thus E m' = ;^2-*'i&r > c^'^jo'{E,) > c^\ Hence W(Gr(f)) > q^ > 0 if t < 2 — a, and we conclude that dim^ Gr(/)= 2-a. • Chapter 4 Self-affine mappings 4.1 Box dimension of self-affine curves Iterated function systems (IFS) provide a very convenient way of representing and reconstructing many fractals that in some way make up of small images of themselves. In what follows these IFSs are used to construct special fractal sets, namely self-affine fractals. This section is concerned with the calculation of the Hausdorff and box dimension of self-affine fractal sets. A fractal set F in E^ is called self-affine iff it is generated by a finite collection of maps Si, i = 1,..., k, of the form S^[x)^Ti[x) + ai (4.1) where T] is a linear transformation on R"，(representable by an n x n matrix) and CLi is a vector in E^. Now, we concern those self-affine curves in R^ plane which in- terpolate some points. Given a collection of points (xo, "o), (^i, Vi), •.., (:n，Vn) G R2, where without loss of generality we take Xo : 0 and Xn : 1. We now construct a self-affine curve through n points in a E^-plane. Choose for each i = 0，• • • ,n-l a linear mapping Si : R^ ^ R^ of the plane of the form 1 \ ( . \ ( \ 1 , \ X I tti 0 X , di ^ + (4.2) \ y J \ bi Ci y 乂 y ) \ ei j 30 The Fractal dimension of the Weierstrass type functions 31 with 0 < CLi < Ci < 1 (4.3) and the further properties that Si{{xo,yo)) = {xi,yi) (4.4) and 5'i((^n, Vn)) = {Xi+l,yi+l) (4.5) Suppose that each Si is a contraction mapping, that is |*^(o:) — *X(y)| < n|x — y for oc,y G R^ where 0 < 7\ < 1. In [8], Hutchinson proved that there is a unique compact set F in the plane with the property that n-l F=[jS,{F) (4.6) i=0 and we call F a self-affine set. 1.8「 ： ;• : •； ： 16-丨 /^fW W^V^ 广 \ n n • '\ …r^/ V 狐丨 1.2-...... f i v/ :..]....:.."r: / ‘ I I \ 1...... f...... 丨...... 丨...... 丨...... 丨...... 一.丨...... fK...... 丨 0.8-....； \ ....' t'； ：、： �.6-.;.... .:. ....： :.....V 0.4 -•t ：；• ； V\ ！ ； ； V 0.2 f •： V1 « Q I I I I I I I I I ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1: Graph of self-afRne mapping with ao = ai — 0.5, bo 二 1, bi 二 一1， co = Ci = 0.6,do = di = 0,eo = 0, ei = 1 For convenience we write [0,1] = I. Conditions (4.4) and (4.5) F contains the points (xo, yo), • • •, (In, Vn)- In fact we will see that F is the graph of a function The Fractal dimension of the Weierstrass type functions 32 f : I ~> R. We now consider the special case in which Xi = i/n for 0 < i < n so that Gi — 1 /n = a and di = ia for 0 < i < n. Let us establish some notation and basic definitions. The full shift on n symbols is the space ^n = {0,..., n - lf = {x - {XI,X2,...) ： Xj e {0, • •., n - 1}}. together with the shift map a : T>n — S„ defined by ¢7((0:1,¾...)) = (¾¾--.). A finite word (x1,x2,..., Xm) is denoted x(m). The composition Sx^ • • • S^^ is written *Sx(m). For a subset E of E^, we define the height of E as E\H = sup{\y-y'\ : {x,y),{x',y') e E} and the width of E are E\w = sup{|x - x'\ : (x,y), {x', y') G E} It is natural to look for a formula for the dimension of self-affine curve. We would hope that the dimension depends on the affine transformations in a rea- sonably simple way, easily expressible in terms of the matrices and vectors that represent the affine transformation. In [9], Bedford found a simple formula for the box dimension of the self-affine curve. To prove that, we need to introduce two basic lemmas first. Lemma 4.1.1 Let Si be an affine mapping defined in (J^.2). For E C I x I, Then, (i) |*Si(E)|v^ = ai\E\w and (ii) \Si{E)\H Proof For (x, y), (xi, yi) G E, Let (u, v) 二 Si(x, y), (ui,Vi) = Si(xi,yi). Then, |ix — Ui\ = ai|a; - Xi| and |” - i*i| = |ci(y — yi) + bi{x — Xi)|. So we have Si{E)lw = ai\E\w and \Si{E)\H < Ci\E\H + |^||^|vy. • The Fractal dimension of the Weierstrass type functions 33 Lemma 4.1.2 Let b = max \bi\, and F be the self-affine set for Si where i = 0, • • .n — 1. IfE C F, we use the notation E"x(m) 二 5'x(m)(^)- Then, m j-l \E^{m)\H < C., ... C,^\E\H + h\E\w Y, a^-^+i(f]〜）. j=l k=l Proof We can proof by induction, Lemma 4.1.1 shows that the statement is true for m = 1. Assume that it is true for m — k, k i-1 \E^{k)\H < Cx, •..c^,\E\H + b\E\wY^a^-'^\Y[c�,.) i=l j—1 When m — A; + 1, this follows directly from the fact that Sx^ {Fx2-Xk+1) — F"xi-xk+i and apply the Lemma 4.1.1. Therefore, ^Xi---Xk + l H ^ ^Xi ^X2---Xfc + 1 H + bxi jEx2---Xk+l W ( k i \ < C^1 c^2 • • • Cxfc+1 \E\H + b\E\w ^ a^~'^^ fJ ^^i ^ba^-^\E\w \ i=l j=2 J k+1 i < Qri ... Qrwi\E\H + b\E\w Y, a^|+2(H C,.) i=l j=l Then, the statement is true for m = k + 1 and the induction is proved. • We can rearrange the expressions for |^x(m)| in Lemma 4.1.2 and apply the condition (4.3) to obtain Ey:{m)lH < Cxi . . . Cx^{\E\H + \E\wdi} where di > 0 is a constant independent of m. By the above two lemmas, we have the following basic estimates: there exists d > 0 such that, for all x G E and m > 0， Ejc{m)lw = ct^^ln^ and E^(m)lH < Cxi"-Cx^{|^|/f+|^kc?l}- (4.7) The Fractal dimension of the Weierstrass type functions 34 Theorem 4.1.3 Let F be the self-affine curve determined by the IFS in (4-2) and (4-3). In the above case, if the points (x^, yi) are not collinear, we have dimBF=l + logy=o\ (4.8) -loga Proof We take qi, q) and q^ to be three non-collinear points chosen from (^¾, yi) {i = 0,"',n), then S^, • --5'x^fe)(i = 1,2,3) is in F^(m). The height of the triangle with these vertices in Fx(m) is at least Cx^ ... Cx^d2 where d2 is the vertical distance from q2 to the segment |^i,g^3]. Taking E = F in the (4.7) then we have the following estimates d'^c� ,.., . C^^ < |^m)U < dc:ri ... Cx^ (4.9) where d — max{|F|i^-+|F|vyc/i, d2^}. For each m, we cover Gr(/) by squares of the form [za^, (z + l)a^] x [ja^, (j + l)a^] where m, i,j e N. If N{m) is the minimum number of such squares (for fixed m) covering F, then [log N{m)]/[-m log a] ^ dims F as m ~> 00 if the limit exists. Now, using our estimates of |^"x(m)|iy in (4.9) and Proposition 2.1.1, we have d-i J2 ^-1 . •. CTma, < N{m) < d ^ c,, ... c,^a-^. x(m) x(m) Rearranging the expression 5Zx(m) Cxi • • • c^^, we have d-'(co + . •. + Cn-ira-^ < N{m) < d{co + ... + c^_O^a"^ and so dimsGr(/) = l + ^^^to. • —loga We remark that Hypothesis 4.1.4 The following conditions are equivalent: (i) The points {xi^ yi) are collinear. The Fractal dimension of the Weierstrass type functions 35 (ii) The eigenspaces of the maps Si coincide. (iii) F is a straight line. Proof If (i) holds then the line containing the points {xi, jji) is invariant under each Si, and thus is the eigenspace for each Si which implies that (ii) holds. If (ii) holds, then letting Fi equal the eigenspace of 5o intersected with [0,1] x R, we have Fi = \J. Si{Fi). By the uniqueness of compact nonempty solutions to this equation, we must have F — Gr(/). In particular F is a straight line, so (ii) implies (iii). Since Gr(/) contains the points (xi, yi), (iii) implies (i) is trivial. This shows that (i), (ii) and (iii) are equivalent. • Corollary 4.1.5 If the points {xi, yi) are not collinear. Let F be the self-affine curve. If (4--2) in the above theorem is replaced by the condition a^ < Ci, then (4-8) still holds. Proof Let E be a subset of F. We rearrange the expressions for |E"x(m)| in Lemma 4.1.2 and apply ai/ci < 1, E^{m)lH < Cxi • -'Ca:^{lElH^mblElw} Then, we have the following estimates which come from (4.9), d~^c^, . •. Cx^ < |^x(m)|^ < rndca：^ ... c^^ where d 二 max{|F|i:f + h\F\w, d2^}. Then, dimBF = l + log�:o \ • —ioga 4.2 DifFerentability of self-affine curves In the above section, we leave out the case that the self-affine curve in [0,1] is differentable. We will consider this case in this section. For simplicity, we set The Fractal dimension of the Weierstrass type functions 36 n = 2 and the the self-affine curve interpolates (0,0),(0.5,1) and (1,0). These three points determines the mappings as / \ / \ X I X 50 = To + bo \yJ \yJ and /� \ /�\ 51 工=Ti � +6i (4.10) \y) \yJ (0.5 0 \ ( 0.5 0 \ ( 0 \ ( 0.5 \ where T� = ，7\ = , h^ = and hi 二 . V 1 Co) \-l Ci) \Oj V 1 / We define a function f whose graph is the self-affine set for Si. We show that this continuous curve is differentable except the dyadic points for Ci < 0.5. First, let Dn be the set of dyadic rationals m!2^ and the set of all dyadic point D = U^iAi. It is easy to see that D is dense in [0,1 . Lemma 4.2.1 Iff is a continuous function in [0,1] satisfy the inequality f{x) - f[y) \ < c\x-y for each x, y G D, then f is a Lipschitz function in [0，1 . Proof For each x, y G [0,1], there exist sequences {x^}, {y^} C D such that Xn, Vn converges to x, y respectively. Since f is continuous, lf{x)-f{y)l = lim lf{xn)-f{yn)l n^oo < lim c\Xn - Vn n—oo < c\x — y\. • We now defines the "depth" of a dyadic number t as the first integer k for which t belongs to Dk. The following lemma is simple to prove by induction. Lemma 4.2.2 Let F be the self-affine curve for Si defined in (4-2) with the corresponding function f. Then for each dyadic number x = O.Xi • • • Xm (base 2) where the depth is m, then we have (x, f{x)) = 5x(0,0) and x = {x1,x2 .. • Xm)- The Fractal dimension of the Weierstrass type functions 37 Lemma 4.2.3 Let F and f as above. For each dyadic number x where the depth is m and each n > m, f{x + l/2^) — f{x) is the left-bottom entry of the matrix Tx' where x' = (x1,a:2? •.., ^m，0,..., 0) where the depth of x! is n. Proof For each dyadic number x where the depth is m and n > m, Lemma 4.2.2 shows that (rr + l/2�/O r+ l/2”) =�(0，0) where y = (xi, X2,. •., Xm, 0，. • •，0，1) and the depth of y is n. By Lemma 4.2.2 and the condition 5i(0,0) 二 5o(l, 0), we have (0： + 1/2”，/(冗 + 1/2-)) = ^...^^5o---^5o(l,0) :=^x'(l,0) where x, = (x1,x2, • •., x^, 0,. •., 0, 0) and the depth of x, is n. Then , (o; + l/2^/(:r + l/2%-(a^/(x)) 二 ^(l,0)-^(0,0) 二 Tx.(l,0) since 5x' (x, y) is a translate of Tx' (x, y). Therefore, f{x + 1/2” — f{x) is the left-bottom entry of the matrix Tx' • Lemma 4.2.4 With the condition in Lemma ^,.2.2, f is a Lipschitz mapping in 0，1:. Proof For each dyadic point x with depth m and for n > m, we have, by Lemma 3.2.3, f{x + l/2-)-f{x) < 0.5“6“0.5“c“". + c,r.c�—An < 6(0.5"-1 + 0.5几-2。+-.. +。冗-1) < 6((l/2)- - c-) _ 1/2-c < r{l/2r The Fractal dimension of the Weierstrass type functions 38 where b = max{|6i|}, r = b/(0.5 — c) and c = max{|cj|}. For each x, y G D, without loss of generality, we assume that x > y and the depth of y is n, so X = y + no/2" for some integer n�. and we have no l/w-/(y)| < ^|/(y + i/2-)-/(y + (i-l)/2-)| i=l no < E"2" i=l < r\x — y By Lemma 4.2.1, f is a Lipschitz function in [0,1] • f is Lipschitz mapping, which means that f is absolutely continuous. In [14], Theorem 7.20 shows that f is differentiable at almost all points of [0,1]. In here we determine which point of f is differentiable. We define a transformation S : R2 ^ R2 by S{X) = So{X)uS,{X) for X C R2. For convenience we write [0,1] = I. Collage theorem[l] shows that S^{I) converge to F in the Hausdorff metric where S^ is the A:-th iterate of S. Denote Fk = S^{I) and a function fk whose graph is S^{I). We now have two lemmas to calculate the left and right derivative of f at point x. Lemma 4.2.5 Ifx is a dyadic point where the depth is k, then fk{x) = f{x). Proof We will prove it by induction. For n = 1, x = 1/2. It is true for n — 1 since f is a self-affine curve that interpolate (0.5,1). Assume that it is true for n — k. When n = k + 1, x is a dyadic point where the depth is k + 1, i.e. X = 0.¾!.. 'ik+i It is easily to see that ik+i = 1, otherwise the depth of x is k. Choose Xi = O.i2 • • 'ik+i' Since the depth of Xi is k and then by inductive hypothesis, we The Fractal dimension of the Weierstrass type functions 39 have (x,/fc+i(x)) = Si,{{xiJk{xi))) =^((xi,/(xi))) =((^,/w)) so, fk(oo) = f{x) and it is true for n = k + 1. The induction is completed. • Lemma 4.2.6 For each x G [0，1]，\fk{x) — /(x)| < rc^ where r is independent of k and c = max{co, Ci}. Proof Since (x, /fc(x)) e Fk and Fk — S(Fk_i), this means that there exists {xi,fk-i{xi)) e Fk-i such that {xJk{x)) = Si{xiJk_i{x^)) (4.11) for some i. Also, we have {xJ{x)) = Si{xiJ{xi)) (4.12) since F is a self-affine curve for Si. By (4.11) and (4.12), we have fk{x) - f{x)l = \biXi + Cifk-l{Xi) + 6i - biXi + Cif{Xi) + 6i =Ci]fk-l{Xi) - f{Xi) < c\fk-l{Xi) - f{Xi) where c = max{co, ci}. Repeat the above process, we have lfk{x)-f{x)l < c'lfo{xo)-f{xo)l < rc^ where xo G [0,1] and r = maXa;^,i] \ f{x)\. • Proposition 4.2.7 With the condition in Lemma J^.2.2, (i) f is differentiable at all nondyadic points. (ii) f is differentiable at dyadic point if and only ifcQ + c： = 0.5 The Fractal dimension of the Weierstrass type functions 40 Proof We first prove (i). Let x be non-dyadic point and write x = O.i1i2 ... ik ... (base 2). For each non-dyadic point x, we define a sequence {yk} to be the first k terms ofx. i.e. yk = ^.i1i2 •. • ik. Then {yk} converges to x. we take a subsequence {ykj} from {yk} such that the last terms of ykj is 1 and the last terms of ykj+i is 0. We define two sequence {aj} and {bj} as aj = ykj + 1/2^^ and bj = y^j — l/2^J. Then,{aj} is decreasing sequence and (¾} is increasing sequence. Since f is continuous, we use the term /(%) - m CLj — X to calculate the right hand derivative of / at x. Since the depth of aj is kj — 1， we have f{dj) - f{x) — fkj-i{aj) - fkj-i{x) fkj-i{x)-f(x) — I • G/ j QC CLj 3^ CL j 00 By Lemma 4.2.6, \fkj-i{x) — f{x)\ is bounded above by rc^J"^ and |a^ — x\ is bounded below by 1/2明，so fkj-l{oo) - f{x) CLj — X converges to 0 since c < 1/2. Since fkj-i is a function whose graph is some segments joining up to form a polygonal curve and aj, x and bj are three collinear points in some segment of Fk, so fkj-i{aj)-fkj-i{x) = fkj-i{aj)-fkj-i{bj) = f{aj) - f{bj) ttj — X CLj — bj CLj — bj Therefore, The right hand derivative of f at oc, f'^{x) is lim fM - /fe) j^oo CLj — bj On the other hand, we use the similar method to calculate the left-hand derivative. We have the left hand derivative of f at x, f'_{x) is lim^M^. j—00 aj — bj The Fractal dimension of the Weierstrass type functions 41 So, / is differentiable at non-dyadic point. Now, we determine the differentability of f at the dyadic points. For each dyadic point x where the depth is n, we use two sequences {yk = x + 1/2^} and {yk = X — 1/2^} to calculate the left hand and right hand derivative of f. By Lemma 3.2.3, we have for each n > m, f{x + l/2^) — f(x) is the left-bottom entry of the matrix Tx^ where Xi = (xi,. •.，Xn, 0,. •., 0) where the depth of x' is k. Then, f(^ + 1/2” — f(x) = (l/2)'-%, + (l/2)'-\b,, + ... + (l/2^—\i •.. Ci“b‘ +¾ •. • c‘bo ((1/2广-时1 + (1/2)^-+¾ + ... + cJ-^i) • Also, f{x) — f{x — l/2^) is the left-bottom entry of the matrix Tx2 where X2 = (xi, •.., Xn-i,0,1. •., 1) where the depth of X2 is k. Then, f{oo) — f{x — 1/2” = {l/2)'-X + (l/2)HQi&2 + ... + (1/2)'-¾, . ..Ct“bo +c“ ... c,J^ ((i/2)^-+i + (1/2)*--+¾ + .. • + ct—^i). Let p(k) = (l/2)'-% + (1/2)^-¾,¾, + • •. + (l/2”-^iQi ... Q,_2a,_1. Since (1/2广-料1 + •.. + ct_^i - 2�0’0广， 1/2 - Co then ( 2^ \ /+W = lim 2^v{k) + 2^a^ ... aJ)^ + Cii ... CiJo zrj^—— fc—00 \l/2 — Co / and /：⑷=^lhn 2'p(k) + 2¾ •.. c‘bo + �...c,J ,Gz2: J The function is differentiable if and only if bi + bo (771^"~" J = bo + 61 (:p7^~~� Vl/2-Ci7 {l/2-CoJ Since bo 二 1 = —6i, the function is differentiable if and only if co + Ci = 1/2. Otherwise, the function is not differentiable at dyadic points. • The Fractal dimension of the Weierstrass type functions 42 4.3 Tagaki function In this section, we show that the graph ofthe Tagaki function defined in (3.13) is actually a self-affine curve when A is an even integer. Tagaki function f is a solution of a functional equation f{x) = X^f{Xx) + g{x) (4.13) where g is the 'zig-zag' function defined in Section 3.4. This is an important property of the Tagaki function and we use this property to obtain the self-affine mappings. Then we use the formula in Section 3.1 to get the box dimension of Tagaki function. Proposition 4.3.1 Let f be the Tagaki function defined in (3.13). If X is an even integer, let k = A/2 and choose for each i 二 0, • • •, A — 1 a linear mapping Sj : R2 _^ R2 (A fx-i 0�,:^ �,A/ - \ Sj 二 + for j = 0, • • •, k - 1 V y / V 2A-1 A- ； V y y V 拟―'7 and 1 . \ ( A-i 0� ( . \ ( ,A- \ Sj — + for j = k,..., A — 1 VW \-2A—i ”) \yJ V2-2jA-iJ Then Gr(/) = U)j5^(Gr(/)) and Gr{f) is a self-affine set Proof For each (u,v) e U^Z^Sj{Gi{f)), there exists {x,v) e Sj{Gr{f)) such that�u v), 二 Si(oc, y). If j < k — 1, / u \ = / A_i 0 \ / X \ + / jX-i \ V ^ y _ V 2A-i 入―。j V y y + V wA-i / _i 丄._i \ a ^x + ja �20!" ½+ a~^y + 2ja_i y The Fractal dimension of the Weierstrass type functions 43 By comparing the entry of the matrix and applying the condition (4.13), we have 7j = 2a'^x 4- X'^y + 2jX'^ =X-y(x)+g{X-'x^2jX-') =X-y{x + 2j)+g{X-'x + 2jX-') =m For j < k — 1, we use similar method to have 2J 二 2A-^ + A""y + 2-2jA-^ =A-y(x)+g(A-'x + 2jA-') =A-y(x + 2j)+g(A-'x + 2jA-') =m Then, Gr(/) is the invariant set for Sj, and Gr(/) is the self-affine set. • Corollary 4.3.2 The box dimension of graph of Tagaki function for even integer A is 1 + logE;: �A-^/logA = 2 - a. 4.4 Hausdorff dimension of self-affine sets Instead of the self-affine curves, we consider the more general self-affines sets in R^. The calculation of Hausdorff dimension for such sets is far more difficult. One of the most general results concerning calculating the Hausdorff dimension and the equality of the Hausdorff and box dimension for such fractals is due to K. Falconer [3]. We first recall that self-affine sets are generated by a finite collection of maps Si, i — 1,..., k, of the form Si{x)=Ti{x) + ai (4.14) where 1\ is a linear transformation on E^. (representable by an n x n matrix) and CLi is a vector in R^. The aim of this section is to determine a number The Fractal dimension of the Weierstrass type functions 44 d{Ti, • • •，Tk) in terms of the linear mapping J\, • • •, T^ defined in (4.14) such that the invariant set F has Hausdorff dimension min{n, d{Ti,.. •, T^)} for almost all a = (ai,...，ak) E M"^ [3]. The expressions of d{Ti,...，T^) is defined as d{Tr,..., n) = inf{5 : [ ^'{Ti,T,,... T,J < 00} (4.15) where the sum is over all finite sequences (zi, •. •，ir) with 1 < ij < k and 0^ are the singular values function of the linear mappings as defined in the following. The main difficulty is as usual, to obtain the lower estimate for the dimension of F, and potential-theoretic method[l] is used to solve it. We assume that the linear mapping T are contracting and non-singular. The singular values a^(! < i < n) of T are the lengths of the principle semiaxes of T{B), where B is the unit ball in M". Equivalently, they are the positive square roots of the eigenvalues of T*T, where T* is the transpose of T. We adopt the convention 1 > ai > «2 > ... > <^n > 0. The singular value function 0^(T) is defined in terms of the singular values of T. For 0 < s < n, define (^s(T) = aia2...c^m-i Lemma 4.4.1 For each s > 0； ¢^ is suhmultiplicative, that is c|>'[TU) < cnT) Proof If 1 < s < n is an integer and E is an 5-dimensi0nal ellipsoid, we have, using (4.17), C'{TU{E)) < WWU(^Ey) < ^S[T)^S[U)CS[E). Thus, (4.18) holds for integral s < n. If m — 1 < s < m where 1 < m < n is an integer, then cf>s(T) = ai...am_icC^+i =(仅1...仅爪_1仅爪广爪+1(“1...«爪—1广-及 _ -^m^j^js-m+l -^m-l^j^ym-s SO (4.18) follows from the integral case. If s > n then the product rule for deter- minants implies that ¢^ is multiplicative. • It is convenient to define some notation which is used in the calculating prod- ucts of matrices by sequences. For r = 0,1,2,...，let Jr = {(zi, • • •, v) : 1 < ij < k}, with Jo consisting the null sequence Joo = {(n,^, • •.) : 1 < ij < k} be the corresponding set of infinite sequences. We abbreviate members of J or Joo as i, and denote the number of terms in the sequence i G J by {i{. Ifi,j G J or j G Joo we write [i,j] for the sequence obtained by juxtaposition. If j 二 [i, i,] for some i', we write i < j. If i,j G Joo then i Aj is the maximal sequence such that both i Aj < i and i Aj < j. We now consider the products of linear mappings J\，. •.，T^ indexed by i G J. We always assume that k > 2. If i = (ii,. • •, v), let Ti = T,�•. 7\. �with T^ = /, the identity mapping. From Lemma 4.4.1， TOj]) < Also, assuming that the T] are non-singular contractions with singular values satisfying 1 > a > ai > • •. > an > b > 0 (4.19) The Fractal dimension of the Weierstrass type functions 46 for 1 < i < k, then for i G J 0i| < 0'(Ti) < a^. If i = (i1,i2,.. •) ^ Joo and a = (ai,..., a^) G R^^. Then by Six = TiX + a^, we have Xi(a) : = lim&i&2...5U0) r~>oo =lim {Ti, + ai,){Ti, + a^J ... (7^ + aJ(0) r—00 =CLii + Ti^cii^ + Ti^Ti^tti^ H (4.20) and we define F(a) = {xi(a) : iGJoo} It is easy to see that the F(a) is the unique self-affine set for the map Si. To define a measure on the sequence space Joo, It is a natural way using the metric d(i,j) = 2-丨1八』丨 for distinct i,j G Joo- This makes Joo into a compact metric space. Writing N\ — {j G Joo : i < j} for i G J, the sets {N� : i G J} form a base of open neighbourhoods for J^o- A set of finite sequences A C J is called a covering set of Joo if for every j G Joo there exists i G A with i < j. Then, we define a measure of Hausdorff type on Joo as follows. Fix s > 0, for each positive integer r let Ml^{E) = inf{5^0s(Ti) : E C UiiV,, |i| > r}. i For each r, A^“ is an outer measure on the subsets of J^. A net measure of Hausdorff type is obtained by letting M'{E) = lim MU(E) r^oo �) Then M^ is an outer measure which restricts to a measure on the Borel subsets of Joo- The measures M.^ are used to define a dimension d[Ti, • • •, T^) in an analogous way to the definition of Hausdorff dimension using Hausdorff measures. The Fractal dimension of the Weierstrass type functions 47 Proposition 4.4.2 The following numbers exist and are all equal: (i) inf{s : M'{Joo) = 0} = sup{s : M'{Joo) = 00}, (ii) the unique s > 0 such that linv_)>oo ^i^j^ ¢^{^1)〜卜—1， 问 inf{s : EiGJ. ^s[Ti) < 00} = sup{5 : EiGj, ^'{Ti) = 00}, We denote the common value by d(Ti,..., Tk). Proof If M^[E) > 0, then M^{E) = 00 if t < 5, so the inf and sup in (i) are equal. Similarly, since 0^(Ti) decreases with s, the two terms in (iii) are the same. To justify (ii), we note that by by Lemma 4.4.1, E 列四=X^X>™ ieJq+r ieJgjeJr < 5^0^(Ti)^(^^(Tj). ieJq jGJr Thus Y2i^j^ 0^(Ti) is a submultiplicative sequence, so by the standard property of such sequences, liniy_^ooEiej^ 0^(Ti)]^/^ exists for each s. If T is a linear transformation on E^ with singular values a^ then for h > 0, ^'{T)a'n < n” < 0奶《?, (4.21) so by (4.19) b^' < (E ^'^'(^i))/(E 糊、< «''• (4.22) iGJr ieJr Thus linv_^oo[X^ie_7r 00 E� ^= EE_ ieJ r=i ieJr converges or diverges according as to whether limHoo[X^ie Finally we show that the value in (ii) < that in (i). Suppose M^{Joo) < 1 for The Fractal dimension of the Weierstrass type functions 48 some s. Then there is a covering set A of Joo such that J2ieA ¢^{^1) ^ 1. Let p = max{|i| : i G A}. Define further covering sets Ar {r > p) by Ar = {ii,...,ig : ij e A, |ii,...，ig| > r and |ii,-",Vi| < ^}- It follows from the submultiplicity of ¢^ that ^0^(Ti-..Ti^Ti) < 0^(Ti-..TiJ^0^(Ti) i6^ ieA < m-.T\qY Using this inductively we get that E^'(^i)^i- ieAr If i G Jr+p then i = [i'，j] for some i' G Ar and |j| < p. Moreover, for each such i' there are at most k^ such j. Since (/>^(Ti) < 0^(Ti/), E 则 < F E 列刀‘）^ kp. ieJr+p i'eAr This is true for all r, so lim7.^ooEiej^ 0^(Ti/)]^/^ < 1. We conclude that the value in (ii)< that in (i). • The following integral is required to estimate in terms of the singular value function. We write Bp for the closed ball in R^ with centre the origin and radius P- Lemma 4.4.3 Let s he non-integral with 0 < s < n. Then there exists a number c < 00，dependent on n, s and p, such that r dx c hp 1^ ~ ¥m for all non-singular linear mapping T on M^. The Fractal dimension of the Weierstrass type functions 49 Proof We have f dx — r dx ]Bp < Tx, Tx >s/2 - Jsp < X, T*Tx >^/2 f f dxi•• • dXn -J ••% (a?x? + ... + alxiy/^ choosing coordinate axes in the directions of the eigenvectors of T*T correspond- ing to the eigenvalues a^,. • •, a^. Substituting yi = aiXi/p gives �-)‘7^4.../pM?^，_ where P is the rectangular parallelepiped {y — (i/i,.. •, yn) ： \yi\ < a^}. Let m be the integer such that m — 1 < s < m. Writing Pi = {yeP:yl + -" + yl<4al} and A = {yeP:yJ + ... + 7/^_i>a^}, we have P C Pi U P2, since |y^| < am on P. Thus transforming the first m coordinates of Pi and the first m — 1 coordinates of P2 into polar coordinates, we get from (4.23) pH(ai...a)/ - < f...f _dyr..dyr^_ P (晰 ^-Us, nooJo C2OLm ... OLn / r~'r^~^dr J Qm < c[am+l • • . Q^nC^m"' + ^2^m • • . ^nQ^m"'"^ for appropriate constants C1,C2, c[, c^ independent of the ai, as required. • For technical reasons it is convenient to assume that ||Ti|| < 1/3 for all i, where || • || is the operator norm induced by the usual Euclidean norm. The Fractal dimension of the Weierstrass type functions 50 Lemma 4.4.4 If s is non-integral with 0 < s < n and ||T^|| < 1/3 (1 < i < k) then there is a number c < oo such that r da � c ^aeBpCRnk |a:i(a)-Xj(a)|^ — 0'(¾) for all distinct i,j G Joo- Proof Let iAj 二 p G J, so that i = [p, i'] and [j — p，j'] with i',j' G Joo- Write p = |p|. Suppose, without loss of generality, that the (distinct) first terms of the sequences i' and j' are 1 and 2 respectively. From (4.20), Xi' (a) - Xy (a) = ai — a) + (T^^+iaip+^ + ^^+1^^+2^^^+3) + . • •) “(¾+! %p+2 + Tjp+iTjp+2^jp+3) + . •.) =ai —tt2 + E(a) where a linear mapping E has norm at most 2 ^^^ rf = 2rj/(1 — rf) with rj = maxi y 二 CLi — tt2 + E{a), a2 = tt2, . • .，CLk 二 CLk which is invertible since I + Ei is invertible, as ||Ei|| < ||E|| < 1. Thus r da _ r � JaeBp 丨而⑷-a:j(a)h — iaGS, |TiAFi'(a) - xy{a)\' /f dyda2 ... dak .../ ~~T. r~ J yeB(2+fc)p;aiG5p ^iAj(y) < ^ -0^(TiAj) by Lemma 4.18, where c is independent of i and j. • The following lemma is an application of the potential theoretic method to obtain the lower bound for the dimension. The Fractal dimension of the Weierstrass type functions 51 Lemma 4.4.5 Suppose that fji is a Borel measure on Joo with 0 < //(Joo) < 00 such that for some s < n [[[^yi)"W <00 (4.24) JjoJjoJaeBp |xi(a)-Xj(a)|^ Then for almost all a G Bp C R^^ (in the sense of nk-dimensional Lebesgue measure), dimijF(a) > s. Proof The function ^(a, i,j) = |xi(a) — xj(a)|"^ is the limit as r ~^ 00 of the increasing sequence of continuous functions min{r, |xi(a) — xj(a)|"^}, and so ^ is a non-negative Borel-measurable function on the product space R^^ x J^ x J^. Thus, applying Fubini's theorem to (4.24), we conclude that for almost all a G Bp r f d^i{i)d^iS) < � Jjoo Jjoo l^i(a)-^(a)|-^ • For each such a we can define a measure jjL on R^ by v{E) = fj,{i : Xi(a) G E}. Since the function Joo ^ 肥 given by i ~> Xi(a) is continuous , it follows that v is a Borel measure on R^ with /x(R") = "(Joo). Moreover, jji is supported by F(a), the image of Joo under this mapping. Thus F(a) supports a mass distribution of finite s-energy, so that dimn F{a) > s, using the proposition 2.3.3. • Theorem 4.4.6 Let Ti, • • •, T^ be linear contracting mapping with ||Tj|| < 1/3 for 1 < i < k and bi, • • •, bk G R^ be vectors. IfF is the affine invariant set satisfying F = uJU(7KF) + ~) (4.25) then, dimi7 F 二 dim^ F = d{Ti, •..，T^) for almost all (ai,. •., ak) G M^" in the sense of nk-dimensional Lebesgue measure. The Fractal dimension of the Weierstrass type functions 52 Proof We now show that dim^ F < d{Ti,.. •, T^) for any ai,..., a^ G R". Let B be a large ball so that Si{B) C B for all i. Given S > 0, we may choose r large enough to get \Si^ ‘ • • Si^{B)\ < 6 for every r-term sequence. Since S^{E) = Uj^Si^ • • • Si^{E) where the union is over the set J^ of all r- term sequences (ii, • • •, ir) with 1 < ij < k, we have F C Uj^5j^ • • • Si^{B). But Sii ... SiT {B) is a translate of the ellipsoid Ti^ ... Ti^ {B) which has princi- pal axes of lengths Ofi|B|, •.., an\B\, where ai,. •., a^ are the singular values of Til • • • ^ir. Thus Si^ • • • SiT is contained in a rectangular parallelepiped P of side length ai\B\,..., an\B\. If 0 < s < n and m is the least integer greater than or equal to s, we may divide P into at most (^)(吟.（^^...一， \OLmJ \OLmJ \ OLm ) cubes of side am\B\ < S. Hence S{^ • • • S{^{B) may be covered by a collection of Ui with \Ui\ < 6^ such that J2m^ < 2^...a^_iair^<|Br i < 2-|W(Th".Tj. Taking such a cover of Si^ ... S�{B) for each (ii,..., v) ^ Jr it follows that 7/^^(F)<2-|5r^0^(T,,...T,J. Jk But r ~> 00 as 6 ^ 0, so W{F) = 0 if s > d{Ti,---,Tk)- Thus diuinF < d{T,r--.n). We now construct a finite measure /i supported by F to obtain the lower esti- mate for diuiH F by using Lemma 4.4.4. Fix p > 0. Let t be non-integral such that 0 < t < min{n, d{Ti, • • •, T^)}, and s such that t < s < min{n, d[Ti, • • •, Tk)}. Then M^{Joo) = 00, so by Falconer [3] Theorem 5.4., there is a compact E C Joo such that 0 < M'{E) < 00 and M'{E n N� )< ci^'{Ti) for i G J. Define a Borel measure )U on Joo by fj.{A) = M^{E n 4), so that /i(M) < ci(/>^(Ti) (iGJ). The Fractal dimension of the Weierstrass type functions 53 Using Lemma 4.4.4， [[[=⑴二, < c [ [ 0^j)�(i)_ Jjoo Jjoo JaeBp ^i(a) - ^(a) ^ Jj^ Jj^ < ^EE^'(^p)"V(^p,i)MA^p.) P^J i^j < cj2J2^W''f^M P^J i^j 00 < cciEE^'(^p)"V^(Tp)MA^p) r=l peJr 00 < cciY^jy�s-MNp� r=l peJr 00 < cci"(E)5^a^-,) r=l < 00 since a < 1 and s > t. Thus, by Lemma 4.4.5, dimi^ F(a) > t for almost all a G Bp. As p is arbitrary, diuin F{a) > t for almost all a G R"^. This is true for all non-integral t < min{n, d(T[, •.., T^；)}. • In Section 4.1, we prove that the box dimension of self-affine curve introduced given in (4.8) is 1 + logX^^i Q/logm. We remark that the dimension formula in (4.8) and (4.15) are actually same in such special case, i.e., d(Tor",Tm-i) = l + ^-^^f^ logm To prove this, we recall that 7] in theorem 4.4.6 is a two by two matrix of the (tti 0 \ • form with ai = l/m. For each i = (zi, •.., v) G J, we estimate the \ bi Ci y lower and upper bound of (/>^(Ti) It is easily seen by induction that ( m-r 0 \ T\:Tir.Tir= 1 , 乂 m'-^bi, + m^-^Ci,bi^ + h c�.. Ci^_^bi^. Ci^ ... c") With the condition (4.3), we observe that the bottom left-hand entry is bounded The Fractal dimension of the Weierstrass type functions 54 by mi-� +m^'^Ci^bi^ H h Q, ... c‘_ib‘ < |mi-r& + m^'^Ci,b H h Q^ .. • Ci^_,b < ((mc)i-r + (mc)2-r + ... + l)a, ... a^_,b < pcn ‘ • ‘ Q.-i (4.26) where b — max |6j|,c = min{cj} > 1/m and p = a/(l — (mc)—i). We define o;i, a2 to be the square roots of the eigenvalues of T^T[ with ai > a2. For 1 < s < 2, (ai 0� we have c/>^(Ti) 二 aiag . Since T\ is a matrix in the form , then the \ hi Ci J singular value ai, a2 is a solution of the equation t^ — y^(ai + Ci)2 + &2^ + tti6i = 0. Now, we assume that ai > 0.2 and claim that there exists constants A^ B > 0 such that Aa{ < a2 < Ba-i By solving the equation, we have 2tti 0；2 二 V(1 + ai/ci)2 + (6i/ci)2 + v^(l — ai/ci)2 + (bi/ay' it is easily to get that a2 is bounded above by 2a{. By the condition (4.26), we have bi Since np 0s(Ti) = cviari = "^, 0^2 The Fractal dimension of the Weierstrass type functions 55 By the claim, we have the following estimate of ^j^j ^^(Ti) E^^Em) Then, we have inf{5 : ^ 0(Ti) < 00} = inf{s : ^ qap^ < 00}. iGJ iGJ Rearranging the sum X^i^j Ci<^i"S we have 00 Y,c,ar^ 二 :EEcz�.c!“l—)"s—i) ieJ r=l \i\=r 00 / I I \ r _ ^ ( Cp H h Cm-l \ 一乙 V m^-1 ) • r=l \ , So, X)iej Ciapi < 00, if and only if Y^I^ Ci/m^"i < 1. Then, We have d(To,.-.，T.i) = l + logE=ic\ logm Chapter 5 Recurrent set and Weierstrass-like functions 5.1 Recurrent curves M. F. Dekking [11] introduced a construction of fractal sets that is primarily based on an interplay between algebra and analysis. Dekking calls them recurrent sets which are limits of a sequence of compact subsets of E^ associated with iterates of a set of given words under an appropriate free semigroup endomorphism. Re- current set is to introduce a method of describing and generating space-filling curves, graphs of nowhere differentiable continuous functions, Cantor type sets and related structures. In Section 5.2, we describe Dekking's fractal formalism that will be used to construct the Weierstrass-like curves[10 . Now we introduce the concept of recurrent set. The main idea of Dekking's construction is to associate the following diagram. G{S) ^ Z^ ~U R^ ~^ M^ Q ^ab ^ab ^9ab >X^ -4"*" -^ -»V G{S) "^ Z^ ^ E^ ~^ M^ This diagram is used to describe how a word in G{S) mapping to a value in E^. 56 The Fractal dimension of the Weierstrass type functions 57 Let S be a finite set, say S = {si, • •., Sn} where n = Card S and let G{S) be the free group generated by S, and 0 an endomorphism on S*. The abelianization of G{S) is the free abelian group Z^; the generators {si,..., Sn} of G{S) correspond to the n-tuples {(1,0, •.., 0)，...，（0,..., 1)} in Z^. Thinking of Z^ as the set of all functions from S to Z; The natural projection homomorphism 7r : G{S) — Z^ can be written as T^{^){y) = ^xy for all X, y G S and extends to all G[S). Here S^y the delta function is defined by f 1 X — y ^xy ~ < 0 otherwise \ i.e. the generator Si is mapped onto the n-tuple (0，• • •，1，0，• • •，0) in Z^ , where 1 is in the z-th position. Oab is the induced endomorphism of 9 and ab stands for abelian. The map i is the canonical embedding of Z^ into E^ = W, where r — Card S. Finally, g is a linear map from E^ onto R^, where 1 < d < r, and Le is an endomorphism of R^ such that the diagram commutes. Let f : G{S) ^ R^ be the composition f 二 g o i o yr. Then / is a homomorphism such that fO = Lef. (5.1) The map Lg : R^ ——> E^, is called a representation of 0. Since g is a linear surjection, Le is uniquely defined up to a linear map in ker(^). Since the kernel of g is a ^a^-invariant subspace ofR^, Lg is determined; in the above sense-by this subspace. Now, we give some examples of the representations of 6 : (i) Peano Curve Let 0 be the endomorphism of G{{a, b}) defined by a — aba, b ^ b~^a~^b~^. Suppose that kei{g) = {0}. Then, and choosing g be the identity function gives Le — Oah- This representation is referred to as the full representation of 6. So The Fractal dimension of the Weierstrass type functions 58 that we have /(a) = (1，0)，f(b) = (0,1). Using f9 = Lef, we have / 2 -1 \ Le = 乂1 -2； (ii) Kiesswetter Curve Let 6 be the endomorphism of G({a, b}) defined by a ~> abbb, b ^ baaa. Let 7T be the projection homomorphism of G{{a, b}) onto Z("》)； so we have the generator a corresponds to the 2-tuples (1,0) in Z^^'^\ the generator b corresponds to the 2-tuples (0,1) in Z^°''^\ Let g be the mapping induced by p((l,0))= (1, l),"((0,1)) 二（1，-1). Then, we have /(a) 二 (1,1), f{b) = (1, -1) and f4 0 \ L0 = . � 0 -2j However, Words should be associated with more general compact sets, such as polygonal lines or squares. We now define a map K[-] : S ^ A^(R^), the nonempty compact subsets of E°^, by requiring for a word G 5*, K[si... sJ = U^,i(X[si] + f(si... 5i_i)) (5.2) (Notation : A + y = {x + y : x e A}, for A C R^,y G R^). An important example of such a map is the polygon map defined by (5.2) and for s G 5, K[s] 二 {a/(s) : 0 < a < 1}. (5.3) Given an endomorphism 6 of S*, with a representation Lg in R^ we always consider maps K[-] : S* ~^ AT(R^), where f is the homomorphism satisfying (5.1). In the following, we prove the existence and uniqueness of the recurrent curves. We need the following lemma, The Fractal dimension of the Weierstrass type functions 59 Lemma 5.1.1 Let L be an endomorphism of E^ with eigenvalues Ai,-..,A& where |Ai| < ... < \Xd\- Then for any X > |Arf|, there exists a C > 0 such that for all v G R^, \L^v\\ < cA^||^;||, n = l,2,... (5.4) Proof The result is an easy consequence of the Spectral Radius Theorem. • We call L to be expansive if all eigenvalues of an endomorphism L of R^ have moduli larger than one, and an endomorphism 0 of G{s) is called null-free if 9{s) + e (the empty word) for all s G S. Theorem 5.1.2 Let 6 be a null-free endomorphism ofS*, and Le an expansive representation of 9. Then, for any nonempty word W, there exists a compact set Ke{W) such that L^^K[9^W] ~> Ke{W) as n ^ oo (5.5) in the Hausdorff metric. The set Kg{W) does not depend on the choice of K[-], and is a curve. We call the set Ke[W) recurrent curves. Proof The Hausdorff metric d{-, •) is defined as, d{A, B) = max{sup inf ||x — y||, sup inf \\x — y||}, xeAV^B xeBV^^ for all A, B G /C(E^). The following two properties of d{-,.) are easily checked. (i) For all A B G /C(R^) and x e R^, d(A + oc, B + x) = d{A, B). (5.6) (ii) For all A1,A2,B1,B2 e /C(E^), d{Ai U A2, Bi U B2) < max{d{Ai,Bi), d{A2, B2)}. (5.7) The Fractal dimension of the Weierstrass type functions 60 By choosing for K[-] to be the polygon map in (5.3), we let V — V1V2. •. Vm be a word. According to (5.2) m K\y] — U(Kbj] + /(^1^2.. -^j-i))- j=i It then follows form (5.1) that we obtain L^^K[9V] from K\y] by replacing each line segment K[vj] + f{v1v2 •.. Vj-i) by the polygon L^^K[6vj] + f{v1v2 •. •巧-i). Therefore, putting for n = 0,1,. •. Kn = LJ^K[6^W], we find by (5.6) and (5.7) that for n = 0,1, • • • d{Kn,Kn+i) = m8^xd{L^^K[s],L^^-^K[Os]). s^S By the assumption on L^, we can choose A such that 1 < A < min{|Ai| : A^ eigenvalue of Le]-. Let c > 0 be as in Lemma 5.1.1 for L^^ and A_i. Then, setting d^ = max{d(K[s], LQ^K[6s]) : s G 5}, we have cpTn,K^i)Wor. Since {K{R^),d{-, •)) is complete, there exists Ke{W) e K,{R^) such that Kn — Ke{W) as n — 00. We now show that Ke{W) is a curve. For each n let O^W = WniWn2 . • • Wn,m{n), and let for j = 1,..., m(n), Inj = [{j - l)/m{n)J/m{n)' . We can choose defining functions kn{t) for the curves Kn satisfying for j = l,...,m(n), kn{hj) = LQ^K[Wnj] + f{Wnl • . • Wn,j_l). The Fractal dimension of the Weierstrass type functions 61 For each t G [0，1], choose u = u{t) in the interval I^j which contains t, such that |&n+i(f) — kn{u)W is minimal. Then, ]K+i{t) - kn{t)l < Wkn+l{t) - ^nM|| + Wkn{u) — kn{t)\ • • 1 < d{L^^K[WnjiLj^-'K[eWnj]) + W^n{^) "�(^)ll < m^xd{L^^K[s], L^^-^K[6s]) + max ||L^"/(5)| < c(c/o + max||/(s)||)A-^. s^S Then kn{t) converge, therefore, uniformly to a continuous k : [0,1] — R^, and it is easy to show that A:([0,1]) = Ke{W). Hence Ko{W) is a curve. It remains to show the independence of Ke{W) of the choice of K[-]. Let K[- and K'['] be two maps satisfying (5.2), and let A,c > 0 be as above. Then, by (5.6) and (5.7), d{LQ^K[6^Wl L^^-^K'[e^W]) < maxd(L^^ir[5], LQ^K'[s]) -ses< maxcA-^d(K[5l,i^'[s]) 、L J, “ . Hence, L^^K'[6^W] also tends to Ke{W). • •*• Figure 3: Peano curve; K^ for n=l,2,3 y\/\/ Figure 4: Kiesswetter curve; Kn for n=l,2,3 The Fractal dimension of the Weierstrass type functions 62 5.2 Recurrent sets We can extend Theorem 5.1.2 slightly. A distinguished role is played by symbols s such that K[s] = (i) e^s G Q* for k > m, (ii) e^s 0 Q* for k > m. where Q* is the semigroup generated by Q. If Q is the set of virtual symbols, then the symbols satisfying (ii) are called essential symbols. Theorem 5.2.1 Let 0 be an endomorphism ofS* and Le an expansive represen- tation of6, f : S* ~> R^ satisfying (5.1), and K[-] : S* ^ JC{W^) a map satisfying (b.2). Let Q = {s G S : K[s] = 0}， and suppose that 0 is Q-stable. Then there exists a nonempty compact set KQ{W) such that L0^K[O^W] ^ Ke[W) as n ^ 00 in the Hausdorff metric, for any word W which contains at least one essential symbol Proof The proof is a minor variation on the proof of Theorem 5.1.2. Let Kn = LQ^K[6^W]. Let m be as in Definition 3.1，and let E 二 {s G S* : e^s 茫 Q*} be the essential symbols. Let, for n > m, 9^(W) = 9^e^-^(W)= 0^{uniUn2 ... Un,i{n))^ and let Vnj = f(�[UnlUn2 . • . Un,j-l))- Then for n = m, m + 1, •.. d(i^n,X,+l) = d [u'j:lL^^K[O^Unj] + ^, u'|:lL^^-'K[6^+'Unj] + Vnj) =d {\Ju^.^EL-e^K[e^Unj\ + Vnj, Uu^.eELo^-'K[0^^'Unj] + Vnj) < umxd{L0^K[e^slL0^-^K[e^+h]), The Fractal dimension of the Weierstrass type functions 63 and the proof is completed as in Theorem 5.2.1. • This theorem extends the recurrent curves to the recurrent sets. A simple example is the Cantor set. Let X = {x, y}, let 0 be defined by X — xyx, y ~> yyy. Let f be the mapping from G{{a, 6}) into M given by f{x) = f{y) = 1. Then, since f{0{x)) = f{x) + f{y) + f{x) 二 3 = /(%)), 0 can be represented by its representation: Lef = S'f. and let K[x] = [0,1], K[y] = 4>. Then, the set of virtual letters Q is {y} and the endomorphism 9 is Q-stable. Therefore, by Theorem 5.2.1，the sequence Kn{x) converges in the Hausdorff metric to a unique compact subset C of R. Figure 5: Cantor set; K^ for n=l,2,3 Note that K,{x) = l/3(i^(x) U [f{x) + K{y)) U {f{xy) + K{x)) -0,1/3] U [2/3,1 K2{y) = [0,1/9] U [2/9,1/3] U [2/3, 5/9] U [8/9,1], Then, Kn[x) = [0, (1/3广]U [2(1/3广,(l/3)"_i] U •.. U [1 - (l/3)", 1]. Hence the limit of Kn is the classical Cantor set. The Fractal dimension of the Weierstrass type functions 64 5.3 Weierstrass-like functions from recurrent sets It provides some of the most basic example of fractal curves. A Weierstrass-like function has the form 00 W{x) = ^X-^^{jy^x) (5.8) n-0 where v > A > 1 and ^ is periodic of period 1. Here Dekking's recurrent set formalism [11] is used to generate some Weierstrass-like functions. Unfortunately, we cannot use recurrent sets to generate the classical Weierstrass function as in Chapter 3. We begin by describing the parts of Dekking's fractal formalism that will be used to construct the Weierstrass-like curves. Let S = {a, h] and let S* be the semigroup generated by S. To words in S* we associate polygonal lines as follows. First let us suppose that f : S* ^ R^ is a homomorphism defined on a and b by f{a) = (1,1) and f{b) = (1, -1). Let K[x] = {t. f{x) I t e [0,1]}, cc = a, b be the associated line segments for a,b and let n K[si... Sn] = U (仲」+ f(Si. . . Si-l)). i=l be the polygonal lines associated with the words in S*. By Theorem 4.1.2, we generate a fractal curve which is the limit Ko{a) of the sequence Kn = L-^{K[e^{a)]) convergence in the HausdorfF metric. Let us define some notation now. For each s G S the s-length of W G S*, denoted by \W\s, is the multiplicity of s in W. We call an endomorphism 6 cautious if, for all s G S whenever 0{s) = Si.. • 5^, we have 1 1 Si... «5山 > -1 and |si... SmU > ^{rn — i + 1) The Fractal dimension of the Weierstrass type functions 65 for 1 < i < m. an endomorphism 0 for two symbols a, b is said to be symmetric if 6{a),0{b) in S* such that if 6{a) = ai... a^ and 0{b) — bi.. • bp then a^ = a if and only if bi — b and a>i = b if and only if bi = a. Now we assume that all substitutions 6 are cautious and symmetric. We also assume that if a — \0{a)\a — \0{a)\b and f3 = \0{a)\a + \0{a)\b then [5 > a > 1. We show that Ke{a) is the graph of a Weierstrass-like function as in (4.9). Using the relation Lf = fO, we get that W"1° )" / Let ko — K[a]. For each n > 0, let k^ ： [0, p^] ~> R be the function whose graph is K[0^{a) . Lemma 5.3.1 There is a continuous function 小:M+ ~> R such that for any n G N， ¢{^) = a'^kn+i{0x) — kn{x) for all X G [0，/^:. Proof The map 办 is continuous since K[0^{a)] is a polygonal line. In order to show that 4> is well defined by the above statement we must first prove some properties of the sequence 0^{a). We claim that for all m > 1, the subword of 9^[a) formed by the first jS^—i letters is equal to 6^~^{a). It is true for m = 1 since 0 is cautious. Assume that ^-(a)=^-i(a)H^^ for some word Wm if 1 < m < k, then 6^+^{a) = 6>m(a)6>(VTm). Note that the length of 6^{a) is |3^. Then,the statement is true for m = k + 1. The induction shows the claim. Now, we have the property that the first |3^ letters of 0^{a) form the word 6^{a). The geometrical interpretation of this is that the sequence of sets K[0^{a)] converges in the following sense: K[e^{a)] = K[6^{a)] n {Or，y) | 0 g x < /T} The Fractal dimension of the Weierstrass type functions 66 for all n < m, or alternatively, kn{x) = km{x) for all x G [0, |3^] if n < m. This shows that a'^kn+i{Px) - kn{x) = a'^km+i{Px) - km{x) for all X G [0, P^] if n < m, and hence that • as defined above is unique. • Remark that 小,though not a periodic function, is unique. Lemma 5.3.2 ^ is a bounded continuous function with the property that for all m G N； either ^{m + x) = 4>{x) or 0(m + x) = -^{x) for all X G [0,1 Proof First note that by the symmetry between a and b, {{x, 0(x)) : X e [0,1]} = {(x, yi - y2) ： (x, yi) G L'^K[Oa], (x, y2) G K[a]} ={{x, -{yi - y2)) ： (x, m) e L-'K[Ob], (x, y2) e K[b]}. Take n large enough that m < /?"，and suppose that 0^{a) = UvW where U and W are words and v is the mth letter of 6^{a). Since K[W1W2] = i^[V^i] U {K[W2] + f{Wi)) for all words Wi and W2, we have {(j; + m,Avj2; + m))| 00 e [0,1]} = K[6^{a)] D {(x + m,|/)| x G [0,1]} =K[v] + f{U). Using the relation Lf = fO, we obtain {(x + m,a'^kn+i{P{x + m))){x G [0,1]} 二 L-iir[r+i(6i)]n{a; + m，W|2;e[0，l]} -L-^K[6v]^f{eU)) =L-'{K[Ov])^f{U)). The Fractal dimension of the Weierstrass type functions 67 Since ^{x + m) = a'^kn+i{|3{x + m)) - kn{x + m) and v is equal to either a or b, for all X G [0,1] either <^(m + x) = ^[x) or 0(m + x) — -^{x). • We can now show that Ke{a) is the graph of a Weierstrass-like function. Theorem 5.3.3 With the above notation, there is a map ^p : [0,1] ^ R given by oo 咖)二;^«-書工） n=0 and Gr((^ + ko) — Ke{a). In particular dirriif Gr(99) = dim^ Ke{a). Proof By Lemma 5.3.1, we get a-^(j){p^x) = QT(^i)AvH>i("n+y) — a-^kn{0^x). we have m ^[x) = lim V[a(^+^)/c,+i(/5"+^) - a-^kn{f5^x)] m—00 ^-^ n-Q =lim o>+i)A;n+i(/f^+iaO — kQ[x). m^oo Therefore, Gr(v? + A;o) 二 lim^_,oo L-^K[6^{a)] = Ke{a). • The Hausdorff dimension of the Weierstrass-like curves is calculated using techniques of random matrix products and is introduced in the section 4 of [10 . Bibliography 1] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applica- tions, New York: J. Wiley k Sons (1990) 2] K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge. 3] K. J. Falconer, The Hausdorff dimension of Self-affine fractals, Math. Proc. Camb. Phil. Soc., 103 (1988), 339-350. 4] K. J. Falconer, The Hausdorff dimension of Self-affine fractals //, Math. Proc. Camb. Phil. Soc.，111 (1992), 169-179. 5] R. D. Mauldin and S. C. Williams, On the Hausdorff dimension of some graphs, Trans. Amer. Math. Soc. 298 (1986)，793-804. 6] G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), 301-325. '7] B. R. Hunt, The Hausdorff dimension of graphs of Weierstrass functions, Proc. Amer. Math. Soc. 126 (1998), 791-800. 8] J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747. 9] T. Bedford, Holder Exponents and Box Dimension for Self-Affine Fractal Functions, Constr. Approx. 5 (1989), 33-48. 68 The Fractal dimension of the Weierstrass type functions 69 10] T. Bedford, On Weierstrass-like functions and random recurrent sets, Math. Proc. Camb. Phil. Soc. 106 (1989), 325-342. 11] M. F. Dekking, Recurrent sets, Adv. in Math. 44 (1982), 78-104. 12] K. Kiesswetter, Ein einfaches Beispiel fiir eine Funktion welche berall stetig iind nicht differenzierbar ist, Math. Phys. Semesterber 13(1966), 216-221. 13] C. A. Rogers, Hausdorff measures, Cambridge University Press, Cambridge. 14] Walter Rudin, Real and Complex analysis, McGraw-Hill international Book. (-u冬^- >L ..., > ,v...... , 锋 h,^& 1 碑 f \7 . 二 ^ .-•.」 " . ^ ' ,,v . * , V. £ • , -I ,,:• . 、 〜 - ^iJ r \ -.1 . . - •. . 身 J 「 . L-. . f , . ,: ::ia./ . , -• s ."• .. ;.>」 ..• • ，. ..- .K_ .、-;:J •= - : = . , , : • . .. . J .. . 5, - — . M..L , -. 、 . . 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